Annals of Pure and Applied Logic 39 (1988) 1-17 North-Holland
SUBSTRUCTURES AND UNIFORM ELIMINATION
FOR p-ADIC FIELDS
Luc BeLAIR*
CNRS, Universite’ de Paris VII, Paris, France
Communicated by Y. Gurevich
Received 30 August 1985; revised 11 February 1987
0. Introduction
Let p be a prime. The theory pCF, of p-adically closed fields of p-rank d was introduced in [21] to study the model theory of finite extension fields (of a given degree d) of the field Qp of p-adic numbers. Among other things Prestel and Roquette generalize Macintyre’s elimination theorem for pCF (d = 1, [19]) but with the addition of constants cl, . . . , cd such that the ci yield a basis for the valuation ring modulo p as a vector space over Fp. A primitive recursive procedure for this elimination has been given in [27] and [13]. For a coun- terexample to see that Macintyre’s language alone doesn’t suffice in general for
d > 1 see [25].
The results of this paper consist of an explicit axiomatization for the universal part of pCFd in the Macintyre-Prestel-Roquette language and a model-theoretic proof of an elimination theorem for Th({Q$, :p prime}).
Section 1 contains basic definitions from [21], to which we refer for further details. We also give basic results we shall appeal to in Section 2.
In Section 2 we give our axiomatization for the universal part of pCFd in the language of elimination to be called &(&) below. This adds a new element in the analogy between the p-adic fields and the real field by giving an exact analog to the notion of ordered field. The basic predicates P, of z&P,), denoting nth-powers, yield for each n a multiplicative subgroup of finite index which we denote by P,‘. In our axiomatization, emphasis is given to the fact that for each of these groups the language of elimination contains closed terms giving a full set of coset representatives. A different axiomatization for (pCF), in Macintyre’s language was obtained independently by E. Robinson [23, 251. We discuss it at
* The results of this paper are based on parts of the author’s doctoral dissertation, Yale University, 1985. I wish to express my thanks to my advisor, Angus Macintyre, for his guidance and encouragement.
This research was supported by NSERC of Canada and Fonds FCAC of QuCbec. Current address of the author: UniversitC du Quebec ?I Montreal, Quebec, Canada. 0168~007WW$3.50 0 1988, Elsevier Science Publishers B.V. (North-Holland)
2 L. B&lair
the end of Section 2. When establishing our axiomatization we get, as a byproduct, first-hand information on the model-theoretic role of coset repre- sentatives for PA in the elimination theory. This can be illustrated by a simple proof of uniqueness of p-adic closures we give in [4]. In a similar way first-hand knowledge of the l-types over models of pCF, can be used to give a treatment of the model theory of pCF, in parallel with Abraham Robinson’s treatment of real closed fields: e.g., proofs are provided in [4] which readily transpose to the case d>l.
Finally, in Section 3, the key argument in the proof of uniqueness of p-adic closures alluded to above is used to give a model-theoretic proof of an elimination theorem for Th({Q, :p prime}). Coset representatives of the PA are again in the foreground. This result might be relevant when looking for some uniformity with respect to p in Denef’s work concerning Poincare series (see [12]). Connection with known elimination theorems is made.
For valuation theory we refer to [22] and for model theory to [7]. We use script letters d, 53, . . . in the standard model-theoretic fashion. We write Rd or R* for the interpretation of the relation symbol R in & and R&(a) for d L R(a). Further notation is listed below.
Notation
L = the language of fields (0, 1, +, -, *, -l), card X = the cardinality of the set X, x = (x1, + * - , x,),
A’ = the group of units of the ring A,
v,(n) = the p-adic valuation of the integer n. If K is a field or a valued field,
char K = the characteristic of K, val K = the value group of K,
v(x) = the value of x for the valuation V, V, = the valuation ring of K,
res K = the residue field of K,
R = the residue of x via -: V,+ res K.
1. Preliminaries
We construe a valued field as a domain equipped with a divisibility relation D(x, y) to be interpreted as v(x) G v(y) (see [20]). The relation D is axiomatized
Substructures and uniform elimination for p-adic fields 3 by the universal axioms
lD(O, l),
D(x, Y) v WY, xl,
q, Y) A WY, z)-, % z>,
D(x, Y) A W’, Y’)~w=‘~ YY’),
w, Y) A ax, Y’)-,W, Y +y’).
Note that one can make sense of a valuation map v for a domain instead of a field. Then, both u and D extend uniquely to similar structures on the field of fractions, namely
v(a/b) = v(a) - v(b) and D(alb, c/d) ~D(ad, cb).
Our language of valued fields, 2, is the language of fields L augmented with a binary predicate D(x, y) to be interpreted as a divisibility relation. We will nonetheless currently refer to the valuation map associated to a given divisibility relation.
Definition 1.1. Let p be a prime. A valued field K of characteristic 0 is a p-valued field if res K is of characteristic p and the dimension of the vector space &I(p)
over FP is finite. If d = dim V,/(p), then K is said to be of p-rank d.
The fields Q and C& with the p-adic valuation are both p-valued fields of p-rank 1. Finite extensions of QP of degree d are p-valued of p-rank d. A p-valued field has finite absolute ramification index (and so has a discrete value group) and finite absolute residue degree (and so has a finite residue field). We refer to these numbers respectively as the p-ramification index, denoted by e, and the p-residue degree, denoted byf. We then have ef = d. A subfield of a p-valued field of a given p-rank need not have same p-rank. However it is the case if one augments the language with constants c2, . . . , cd and interpret 1, cz, , . . , cd as giving a basis for V/(p) over lFP in any p-valued field of p-rank d. We denote by zd this extension of 2. We sometimes let c1 stand for the constant 1 in &. Note -re, = _Y.
1.2. Let n(w) denote the following formula of &
D(1, w)A~D(w,
l)AD(~,p)A~[D($ljcj,
l)VD(w,
$~jCj):O~~j<P}*Lemma 1.3. In a p-valuedfield of p-rank d an element w is a prime element if and only if n(w) holds.
Proof. Suppose n(w) holds and let v(y) >O. There are 0~ lj <p such that
4 L. B&lair
v(C ljcj) <v(p) and v(y) = v(C liCj) 3 v(w). Hence n(w) implies that V(W) is the least positive element of the value group, as wanted. The converse is clear. Cl
We record a strong (but equivalent) form of Hensel’s lemma.
Lemma 1.4. Let K be an henselian valued field and
f E
VJX]. Zf there is some a E V, such that v(f (a)) > 2v(f ‘(a)), then there is b E V, such that f(b) = 0 and v(b - a) > v(f ‘(a)).Lemma 1.5. Let K be an henselian p-valued field with p-ramification index e and
E E N such that p % E and E > e. Then the valuation ring of K is algebraically definable by x E V, iff 1 +px” is an e-th power.
Proof. Necessity follows from Hensel’s lemma. For sufficiency, if 1 +px” = y”
and v(x) < 0, then v(y) < 0 and v(p) = ~v(yx-~), contradicting the choice of E. 0
It follows that being an henselian p-valued field of a given p-rank can be axiomatized in the language of rings since we have D(x, y) iff D(1, yx-‘) iff 1 + p( yx-‘)’ is an sth power iff xE + py E is an E-th power. We now state some key facts. Let K be an henselian p-valued field of p-rank d, Y c V, be a complete set of representatives for res K, and JC E K be a prime element. Let P(d, n) = 2dv,,(n).
Fact 1.6. For all n E N, any x E V, has a finite expansion x=$y,n’+x’ with Y,EY and v(x’)>nv(n).
0
Fact 1.7. Zf x E K and v(x) = 0, then x is an n-th power if and only if v(x - An) > 2v(n) for some A = CgCd3”’ y& with yi E Y and v(A) = 0.
Fact 1.8. Zf x E K and v(x) = 0, then )3x is an n-th power for some A as above.
Fact 1.6 is true for any valued field with a discrete value group. The two others are combinations of 1.6 with Lemma 1.4: e.g. in 1.8 apply 1.6 to x-l and 1.4 to P-AX.
Definition 1.9. We denote by pCFd the theory of henselian p-valued fields of p-rank d with value group a Z-group. We write pCF when d = 1. Models of pCFd are called p-adically closed fields of p-rank d.
Substructures and uniform elimination for p-adic fields 5 Lemma 1.10. In the theory pCF, the multiplicative subgroup P;, of non-zero n-th
powers has finite index with coset representatives among the IZJC’, 0 s r < n and il like in Fact 1.7.
Proof. Let K LpCFd, x E K. We have v(x) = nv(y) + rv(;rd) for some y E K and 0 s r < n since val K is a Z-group. Hence v(xnVryMn) = 0 and the result follows by Fact 1.8. 0
Definition 1.11. We denote by L&(PW) the language Zd augmented with unary predicates P,, for n 3 2, each P, to be interpreted as the n-th powers in models of
PCF~-
Prestel and Roquette showed that pCF, admits elimination of quantifiers in
&(P,). Finally let us remark that if K is a p-valued field of p-rank d, then res K is a quotient ring of V,/(p) so that there is always a complete set of representatives for res K among the C ZiCj, 0 s lj <p.
2. The universal theory of p-adic fields
Let p be a fixed prime throughout. We give here an explicit axiomatization for the universal part of the theory pCFd in the language L&(P,). Recall that in
Z&P,) substructures of models of pCF, are p-valued fields of p-rank d.
It is convenient to establish the following notation.
Definition 2.1. For integers n 3 2, d > 1 let B(n) = 2v,(n), P(4 n) =2$(n), n,={nE~:l~:A.~pS(“)+1,p~~}, R,={rEN:OGr<n}, A,={l~~:I=~p’,~~A,,r~R,}, N,, = {i E N :0 6 i <pp@)+l, i is an n-th power modpS@‘)+l}, &,n(%, * . - , “fl(c+z), w) = a,+ a’1w + * - ’ + ~fi(d,n)WB(d’n), Ed = {I, + i2C2 + * “+l,C,:o~~~<p},
n(w) := D(1, W A ) -‘D(W, 1) A D(W, p) A A{o(t, 1) V D(W, Z): Z E Ed},
U(x) := D(1, X) A D(X, 1).
We point Out that Ed iS a finite Set Of closed temS in & and n(W), U(X) are quantifier free formulas in &, 3 respectively. As we saw previously n(w) defines a prime element in a p-valued field of p-rank d if c2, . . . , cd are correctly
6 L. Bdair
interpreted. In a valued field with divisibility relation D, U(X) says that x is a unit in the valuation ring.
Let Tr = T be the following theory.
Axiom 1. Axioms for a field of characteristic 0. Axiom 2. Axioms for a p-valuation.
2.1. Axioms for a divisibility relation D(x, y). 2.2. 1D(p, l), D(x, 1) v D(p, X).
2.3. D(1, x)-, V{D(p, x - i) :0 s i <p}.
Axiom 3. Explicit definition of P, for units of the valuation ring. U(X)-* [P,(x) t, V{D(ps(“)+‘, x - i) :i E ZV”}].
Axiom 4. Behaviour of the P,. 4.1. P,(x”). 4.2. P,(X) A P,(Y)+c(xY). 4.3. P,(x)+ P,(x_‘). 4.4. P,,(x)* P,(x). 4.5. P,(x)-+ P,,(x”). 4.6. ~{Z’&‘x): 3L E A,, r E R,}. Axiom 5. P,(x)+ D(z”x, 1) v D(p”, Yx).
The first two axioms are self explanatory in view of Section 1. Axiom 3 is the result we proved about the definability of n-th powers (Fact 1.7) solely in terms of the valuation. Axioms 4.1, 2, 3, 6 say that P;, is a subgroup of finite index of the multiplicative group, with coset representatives in A,. Axiom 5 keeps the P, consistent with the (lack of) ramification. The following lemma is crucial.
Lemma 2.2. Let q, n(l), . . . , n(k) 3 2 and A = Anclj X * . - X Anckj. We have T b P&x)- ,y’ A {Pn(,~4(&), &,(K’) : 1 ss, t, u s k 44 1 n(u)>.
Proof. It suffices to see that for any II 3 2 there is 1, E A, such that P,,(l$x). Indeed, taking IZ = lcm(n(s)) and 1, E A,,, such that P,,,(I,f;‘) (Axiom 4.6), we easily get Pncs,,(l,4x) (Ax iom 4); moreover if n(t) ( n(u), then Pn~r~(ZuZ;l) by Axiom 4.4 so that Pnc,,(l,l;l) (Axiom 4).
Now let Ap’ E A,, be such that Pn,(Qfx). By Axiom 4 we have P,(ilp’x) and P&p’). Axiom 5 and v,(A) = 0 imply that I = qr’ for some 0 6 r’ <n, so that P,(A). Now A is an integer and u,(n) = 0, so by Axiom 3 and since Q is dense in its henselization with respect to uP (or alternatively, by arguing as if to ‘construct’ a q-th root of Iz in Q, but using only a finite number of steps (approximations)),
Substructures and uniform elimination for p-adic fields 7 there is an integer A, E A, such that A-‘A_,4 satisfies the conditions in Axiom 3 in order that P,,(A-ld,Q). It follows that P,,(&) for Z,, = &p” E A,, as wanted. Cl
To establish our axiomatization we have to show that we can embed a given model AZZ of T in a p-adically closed field (of p-rank 1) A. The process of going from & to .A involves extending AZ to larger Z’(P,)-structures by adding an n-th root to each element a E & for which P,(a) holds. Let yq = a in a p-adically closed field (of p-rank 1) and let l,,, n 22, be integers such that P,(f,y) holds. Then P,,(&), and the statements of Lemma 2.2 hold uniformly for the 1,. The
sequence (1,J tells us in which coset of PI, y lies for each 12 and, as we shall see later, this information determines the type of y, at least as far as the P,, are concerned. Lemma 2.2 gives consistency conditions for a E SIZ such that P,(u)
holds in order to keep available to us (in some elementary extension) the P,-type of some q-th root of a.
Let Td be the following theory, d a fixed integer d > 1. Axiom Id. Axioms for a field of characteristic 0.
Axiom 2d. Axioms for a p-valuation of p-rank d.
2. Id. Axioms for a divisibility relation D(x, y).
2.2d. lD(p, l), D(1, I$, i = 2, . . . , d. 2.3d. D(1, x)-+ V{D(p, x - t) : z E Ed}. 2.4d. lD(p, II + - - * + l&d), 0 G Zi <p not all 0.
Axiom 3d Explicit definition of P,, for the units of the valuation ring. U(x) * Jr(w)+ [P,(x) w VUW’, x - (&f,“(~J w))“)
A lD(X - (g&Z, W)n, n’) : z E Epn)+l}]. Axiom 4-d. Behaviour of the P,,.
4.ld to 4.5d are the same as 4.1 to 4.5.
4.6d. n(w)-+ V{P,,(g&, w)w’x) A U(g,,,(z, w)) : r E R,, z E Es(“)+*}. Axiom 5d. P,(x) A z(w)--, D(z”x, 1) v D(w”, 2”~).
The analogy between Td and TI is clear anough. Note that T, is universal and it is straightforward to verify pCFd != Td. The following is the analog of Lemma 2.2. Lemma 2.3. Let d > 1 and consider q, n(l), . . . , n(k) 2 2, R = R+) x - . - x R n(k) und E = Ez@.‘@))+* x . . . x EgCdjnCk))+‘. For t E E and 1 s i s k let c be in E$(dzn(i))+l and denote the i-th component of t: Then we have
Td b-P,(X) A n(W)-+
(z,r)xXR
A
{Pn(s)q(kd,n(s)(?ww)w”=‘>qx),
ukd,n(s)(t,,w))~
8 L. Bd’air
Proof. The proof is the same as in Lemma 2.2, but using the henselization of the field generated by the constants of 9d (Fact 1.6) instead of the henselization of (a, u,). q
We shall see that every model of Td can be embedded in a model of pCF,, thus showing Td = (pCF&.
Theorem 2.4. Let Oe k Td. Then s4 4 JI for some JU bpCFd.
Lemma 2.5. We have T kP,,(x) A D(x, y) A D(y, x)+ V {P,(Ay):A. E A,}, and for d > 1,
G UP, A D(x, Y) A D(Y, x) A JG(w)
--, // {P,(g,,,(r, W)Y) A u(&,(r, W): * E @d’“)+l].
Proof. Use Axiom 4.6d for x-‘y and Axiom 5d to see that r = 0. 0
Lemma 2.6. Let d I= Td and & kpCFd such that A is a Zd-substructure of M. If Pf c P$ for all n, then d E JU.
Proof. We have Pt E Pr’ and [A ‘: Pt.1 = [M’: Pr’] is finite, and Pt’ and P,“.
have the same coset representatives already in A ‘. 0
Any p-valued field of p-rank d, in particular a model of Td, can be embedded as a valued field in a model of pCFd, see e.g. [8] for a suitable Zorn’s lemma argument. We are now reduced to show that given a model .# of Td we can start adding the required n-th roots and stay inside a model of Td. First we reduce to the case when ti is henselian.
Lemma 2.7. Assume & k Td and B is an immediate henselian valued field extension of A. Then B can be expanded to a model 93 2 d of Td.
Proof. Put I$ = cf. Since B/A is immediate, B is a p-valued field of p-rank d
with the same p-ramification index and p-residue degree and 1, c2, . . . , cd still form a basis for Vg/(p). We do the case d = 1 for definiteness. If x E B, there is some y E A such that v(xy) = 0. Since B is henselian p-valued we get A,xy = b”
for some A, E A, and b E B. Define P:(x) iff Pf(A,y).
(i) This definition is independent of the y and A, chosen. Indeed suppose v(xy) = 0, ~(xy’) = 0, Axy, A’xy’ are n-th powers in B, Pt()Ly) and h, 3L’, y, y’ as above. Then v(A-‘~-~A’~‘) = 0 and X’Y-~~L’Y’ satisfies the residue condition in Axiom 3 so that Pt(A-‘y-lA’y’) and Pt(J.‘y’) by Axiom 4.
(ii) P,” fl A = Pt, for all n: easily seen from (i) and using T. (iii) We verify the remaining axioms for 9 = (B, P,” : n 3 2).
Substructures and uniform elimination for p-adic jields 9
Axiom 3. Use Fact 1.7 and similar axiom in A.
Axiom 4.1. Use y ” for xn if 21 (xy ) = 0.
Axioms 4.2-4.5. For example we verify Axiom 4.2. Suppose Pf(xl), Pf(xJ, u(x,y,) = 0, &xiy, is an n-th power and Pt(&y,). Then u(x1x2y1y2) = 0. Let A be chosen such that Ail;‘&’ is an n-th power, so Pt(n&‘&‘) (Axiom 3). Thus Izx1x2yly2 is an n-th power and Pt(Ay,yJ (Axiom 4.2 in A) whence Pf(xlxJ.
Axiom 4.6. Let x # 0, v(xy) = 0, y EA. It is not hard to get P;;‘(Q-‘y) for suitable A, r, v(n) = 0 (cf. Axiom 4 and Lemma 2.5 in A). We have v(hy) = 0 so n’A~y is an n-th power for some suitable A’. Hence ~(A’p’xp-‘y) = 0, hk’p’xp-‘y is an n-th power and e(k~-~y). So Pf()L’p’;r).
Axiom 5. Let x, z E B be such that P,“(x). So for some y EA and il E An, 3Jcy is an n-th power and Pe(ily). Hence P:(d-‘y-l), V(X) = v(il-‘y-l). The conclusion follows because the axiom is true in A. 0
Note that we can drop the henselian assumption from the preceding lemma. First use Axiom 3d to define P,, for the units of the valuation ring. Then there is A, such that il,,xy satisfies Axiom 3d, etc.
Lemma 2.8. Suppose that for all .c& k Td and all prime q and a E A such that PC(a) we can embed d in a model of Td where a is a q-th power. Then the same is true for all natural numbers n.
Proof. By induction on n and using Lemma 2.5 and Axiom 4.6d we can go to a model % E Td with b E B such that P,(ab-“) and v(ab-“) = 0. By the previous lemma we can assume B henselian whence the result by Axiom 3d and Fact 1.7. cl
Remark that as we just saw above, if & k Td is henselian and P:(a), then a is an n-th power iff v(a) is divisible by n.
Lemma 2.9. Let d k Td, q prime, a E A such that P:(a). We can embed & in a model of Td where a is a q-th power.
Proof. Relying on the previous work we can assume A is henselian, v(a) > 0 and
q + v(a) in val A.
By Lemma 2.3 (2.2) and compactness, there is 53 Z= .s$ on E B, n = 2, 3, . . . such that P&(pza) and P,“(p,p;A) for all n, m. Then a is not a q-th powery B
and X4-a is irreducible over B. Note that B is also henselian p-valued of p-rank d.
Consider the valued field extension B(a)/B, a4 = a. It has degree q. Now q t v(a) in val B and q is prime, so for any x E B(a), v(x) = v(ba’) for some
b E B and 0 < i <q. This together with Axiom 5d ensures that the p-ramification index does not increase in B(a). These considerations and v(o) > 0 imply also that the residue field does not extend. In fact, if u(C bia’) 2 0, then V(bi&) 2 0
10 L. Blair
for all i and u(b,$) > 0 for i > 0. Thus (B(a), cf) is a p-valued field of p-rank d and is henselian.
Let Y = {ba’: b E B, 6 #O, 0 s i <q} and define P,‘(ba’) iff Pf(bp;‘). We show that (Y, cf, P,‘) satisfies the axioms of Td concerning the P,, and then use those P,’ to expand B(a). Notice that if b E B, 1 E N, I = kq + i, 0 G i <q, then Pfl(bp;‘) iff Pz(Z~p;~~p;‘) iff Pfl(bakp;‘) (as P~(p$z)) iff PT(ba’). Also P,’ extends Pf, i.e., P,‘fl B = P,“.
Axiom 3d. Notice that for y E Y, v(y) = 0 iff y E B.
Axiom 4d. For example Axiom 4.3d. Let y = b& E Y be such that P,‘(y), i.e.,
Pf(bp;‘). Then Pf(b-‘pf,), y-l = b-la-‘~q-‘. Since P,“(pza) we get
Pf(b-‘~-~p’,-~), i.e., Pz(y-‘). Use the compatibility PfT(pnp$ in Axioms 4.4d and 4.5d.
Axiom 5d. We show that if y E Y, z, w E B(cY), n(w) and P,‘(y), then v(z”y) G 0 or v(z”y) 2 nv(w). Suppose y = bai and Pt(bp;j). We have V(Z) =
v(b’c~‘)forsomeb’~BandO~i<q. Nowif
O< v(z”y) = v(b’“) + G(a) + v(b) +jv(cu) <m(w), then
(*) 0 < nqv(b’) + v(u”‘b%j) < nqv(w).
But P,“(u), Pfib(p$z), Pf(bp;j) yield P~q(u”‘bqu’) so that (*) would contradict the conclusion of the similar axiom in B for the element u”‘bqui.
So we have established (Y, c;, P,‘) F “relevant axioms of Td” and P,‘n B = P,“. Let M = B(a). For x E M there is y E Y such that v(xy) = 0, and since M is henselian p-valued of p-rank d there is some A E B, v(A) = 0, such that hry is an n-th power in M. Define P,“(x) iff PT(Ay). We can then proceed as in Lemma 2.7 to show that (M, c?, Pf) k Td and Pfl” n Y = PT. This completes the proof. 0 Proof of Theprem 2.4. Use Lemma 2.8, Lemma 2.9 and a standard model- theoretic argument to embed d in a model of Td where every a E Pt is an n-th power. Embed this model (as a valued field) in a model of pCF, and conclude by
Lemma 2.6. q
Remark that since pCFd k D(x, y) ++ P&x’ + py “) where E can be any positive integer prime to p and larger than d, the above theory immediately gives also an axiomatization of (pCFd)v in L(q, P,), i.e., the language obtained when we drop the divisibility relation symbol D.
We now compare our axiomatization with that of [25]. Robinson’s axiomatiza- tion of (pCF)v in X(Pw) relies on the fact that the group PA of n-th powers is ‘effectively open’, namely there is an integer r,, such that if X, y # 0, x E PA and V(X - y) > V(X) + r,v(p), then y E PA. It is clear how to relate this to Hensel’s Lemma. He also includes the diagram of Q (in d;p(P,)) but it is not hard to see that it is contained in our Axiom 3 and Axiom 5. The other axioms are mainly the
Substructures and uniform elimination for p-adic fields 11
same as in Axiom 4. So the essential difference lies in Axiom 3. In fact, “,;I is effectively open” follows directly from Axiom 3 and conversely; they are interchangeable. To establish his axiomatization Robinson needs to know explicitly the (number of) n-th roots of 1 in pCF, which can be done easily because of lack of ramification. The proof is also tied to the rigidity of p-adic closures, which is established at the same time. It is clear that Axiom 3d is interchangeable as well with a suitable version of “Pi is effectively open” in
PCFd.
Any axiomatization of (pCF,)v in &(P,) relates to the description of the points of the p-adic spectrum associated to each completion of pCFd, see [24], [3], [26], [5]. Along these lines it is clear that Lemma 2.3 is closely related to the description of Briicker and Schinke [5], via their neat use of 5 (L ‘/L ‘“), L/Q,, a fixed finite extension.
3. Uniformity of elimination
We now state an elimination theorem for the Q,, when p varies through the primes, namely for the theory Th({Q, :p prime}). We take into account the residue theory Th({ 5P :p prime}), which contains a theory of pseudo-finite fields. A reasonable elimination theorem was shown to hold for those by Kiefe [17], building on the work of Ax [l]. As a theory of valued fields our theory has two kinds of models:
(1) those with a residue field of non-zero characteristic p, which are p-adically closed of p-rank 1;
(2) those of equal characteristic zero, which are henselian valued in a Z-group with residue field a pseudo-finite field.
Having Shoenfield’s criterion for elimination of quantifiers (later E.Q.) in mind, we can split the analysis into those two possibilities. The first one is handled by Macintyre’s Theorem. The second one can be taken care of by the Theorem 5 in [9], or Corollaire 2.21 in [ll], or Theorem 4.12 in [27]. With techniques of Delon [ll] we give here an independent proof based on the key argument in our proof of uniqueness of p-adic closures mentioned in Section 0. The global elimination theorem we get for Th({QP :p prime}) can also be deduced from the Main Theorem 4.3 in [27]. We discuss this more precisely below. Similar questions are treated in an unpublished paper of Fried [14], but in the very different framework of [16] (see also [15]).
We refer to [l] for pseudo-finite fields, e.g., the first-order axiomatization we implicitly use. We need some preliminary results.
Lemma 3.1. Let n be a fined positive integer. There is a uniform bound 6(n) for the index [6& : P;l] when p varies through the primes.
12 L. B&lair
Proof. For all p > n, (n, p) = 1 and using Hensel’s Lemma it is not difficult to see that [Q; : PA] = n[F; : PA] = n(gcd(n, p - 1)). 0
Lemma 3.2. The theory Th({FP :p prime}) admits elimination of quantifiers in the language of fields augmented with nary predicates Sol,, interpreted as
Sol&,, . . . , x,) * 3y (y” + x1 y”-l + * * * + x, = 0) in the theory. Proof. See [17]. 0
Lemma 3.3. Let m, n, d be positive integers. We can find a positive integer f3(m, n, d) with the following property. Zf k is a finite field with card k >
P(m, n, d), then for all fit . . . , fm E k[X1, . . . , X,,], deg& G d, such that the ideal Z=(fl,. . . , fm) in k[X] is absolutely irreducible, the variety defined by Z has a k-point.
Proof. See [l , Section 81. Cl
Lemma 3.4. For any prime p, JU !=pCF, x E M, we have v(x) 3 0 iff P,(l +px’) or P,(l +p(1 +x)“).
Proof. (Cf. Axiom 3 in Section 2.) Suppose v(x) 2 0. If p # 2, then readily P,(l +px’). If p = 2, then v(1 - (1 + 2~)‘) 3 3 when v(x) >O, and v(1 - (1 + 2(1+ x)“)) 2 3 when v(x) = 0, so that accordingly P,(l + 2x2) or P,(l + 2(1 +x)“). Suppose 1 +px* = y* and v(x) < 0. Then v(y) < 0 and 2 1 v(p) which is absurd. Similarly if 1 -t p(1 + x)” = y* and v(x) < 0. 0
Let Z” be the language whose vocabulary consists of the vocabulary of Z(Po), a new constant t, and for n 3 2, a n-ary predicate Sol, and new constants
U l&l> * * * 7 u,,~,(,,) where 6’(n) = n-‘6(n), 6(n) given by Lemma 3.1. We first give
an explicit axiomatization for Th( { QP : p prime}).
Let T’ be the theory in 9’ consisting of the following axioms: (1) Axioms for an henselian valued field of characteristic 0. (2) The value group is a Z-group with unit v(t).
(3) p,(x) t, 3y (y” =x)9 D(“n,jT l) h D(l, 4z.j)*
(4) S&(x,, . . * , xn) t, A D(17 xj) h 3Y (D(l, Y)
A D(t, y” + x1yn-l+ . * * +x,)).
(5) If the residue field has characteristic p # 0, then it hasp elements and t = p. (6) D(x, y) f, P,(x” + ty’) v P,(x’ + t(x - y)*).
(7) V {P,(u,,~~~x): 1 <j < 6’(n), 0 S r <n}.
(8) If the residue field has more than /3(m, n, d) elements, P(m, n, d) from Lemma 3.3, and fl(X,, . . . , X,), . . . , f,(X1, . . . , X,) are polynomials of degree
<d over the valuation ring such that their image $ under the residue map generate an absolutely irreducible ideal over the residue field, then the $ have a common zero in the residue field.
Substructures and uniform elimination for p-adic fields 13
(9) The residue field is quasi-finite, i.e., it is perfect and has a unique extension of each degree.
Proposition 3.5. The theories T’ and Th( { Q, : p prime}) have the same models. Proof. In view of our discussion it is clear that Th({Qp :p prime}) k T’. On the other hand, let A4 != T’. If char res M is p #0, then M bpCF, so M b Th({Q, :p prime}). If char res M is 0, then M is henselian of equal character- istic 0, val M is a Z-group, and res M is a pseudo-finite field of characteristic 0. By [l] there is some ultrafilter 9 on the set of primes such that res M = II FJ9.
It follows by Ax-Kochen-Ershov that M = II U&/S as valued fields. So
M kTh({Q, :p prime}). 0
We need the next two lemmas in the proof of the elimination theorem.
Lemma 3.6. Let K be an henselian valued field of equal characteristic 0 and K0 be a subfield of K. The following are equivalent.
(i) The residue map induces an isomorphism of K0 onto res K.
(ii) K,, is a maximal trivially valued subfield of K. Proof. See, e.g., [18, Lemma 81. Cl
Lemma 3.7. Let E G F be valued fields, F0 E F be such that the residue map induces an isomorphism a of F0 onto res F. Suppose that E0 = cu-‘[res E] is
contained in E and let K0 be such that E,, E K0 c FO. Then
(i) E and F0 are linearly disjoint over EO.
(ii) Any x E E[K,] can be written x = c; eiki, e, E E, ki E K,, and v(ei) < v(ej) if i<j.
(iii) val EK, = val E and res EK, = res K,,. Proof. See [ll, Proposition 2.151. Cl
Theorem 3.8. The theory T’ admits elimination of quantifiers in 9’.
Proof. We use Shoenfield’s criterion. Let A&, A& be models of T’, &i c J&, f : dl 7 d2 such that card MI = w and J% is w,-saturated. We have to see that f
extends to an embedding A& c, 4.
Case 1: The field res AI has characteristic p # 0. Then so do res Mi and the Mi
are p-adically closed of p-rank 1. We get the desired extension by Macintyre’s Theorem.
Case 2: The field res AI has characteristic 0. Then the Mi are henselian valued fields of equal characteristic 0 valued in a Z-group with pseudo-finite residue fields. We argue in the style of [ 111 to reduce to res AI = res MI; then use our argument to reduce further to val AI being pure in val MI, and finally close the case with Ax-Kochen-Ershov.
14 L. Bdair
(2.1) We may assume Ai is henselian. Let A! be the henselization of A, in Mi. Then f extends to an isomorphism of valued fields f h between the A:. This f h is compatible with the Sol, because those are determined at the level of residue fields and At/A, is an immediate extension. For the P,, let y E A:, v(y) 5 0.
There is x E Al such that v(x) = v(y) < v (x - y) and by Hensel’s Lemma we have x E M;” iff y E Min. Clearly such a configuration transfers to M2 via f” (and vice versa) and the desired conclusion follows.
(2.2) We may assume res A, = res Ml. First, observe that with the interpreta- tion of Sol, in Mi, we can define in a natural way predicates Sol, in res Mi which coincide with the Sol, of Lemma 3.2. By Lemma 3.2 there exists
g:(resM,,S&)r(resM,,S&,)
extending the map induced by f on the residue fields (res Ai, Sol,) considered as substructures. By Lemma’3.6 and (2.1), let A0 G Al be such that the residue map induces an isomorphism of A0 onto res Al, and Ni E Mi with the similar property such that A0 c Nl. Let oi : Ni * Mi be the inverse to the residue map. By Lemma 3.7, Al and Nr are linearly disjoint over Ao, valAINI = val A,, resAINI = res NI = res Ml and any z E Al[Nl] can be written z = C X,yi with Xi E Al, yi E Nl
and V(Xi) <u(xj) if i <j. We can thus define a map f’:Al(Nl)+A,(~.ga;l[Nl])
which extends f and is an isomorphism of valued fields. To see that f’ is an Y-isomorphism, first remark that the Sol, are immediately taken care of because of g. For the P,,, let z = C X,yi with Xi, yi as above. Then V(Z) = V(XjYj) for some j and Hensel’s Lemma implies z E Mi” iff XjYj E Mi”. Let u = u,,k be such that Uyj E Min. Then z E Mi” iff XjYj E Mi” iff XjU-r E Mi” iff f(xj)U-1 E Mi” iff
f (xj)f ‘(yj) E Mi” iff f’(z) E M;“. (Note that the analytic configuration of Z, Xi, yj
carries over to M,.)
(2.3) We may assume valA, is pure in val Ml. First note that we need only worry about prime numbers. Let q be a prime. By axiom (7) of T’ it suffices to add a q-th root to any a, eA1 such that Ml ‘F P,(ul) but q c v(al) in val A,, and extend f accordingly. So let al E Al be as above such that, w.l.o.g, v(al) > 0. Let y, E Ml, p,, = u,,jt’ such that y: = al and onyl E Mi”. Let u2 = f (aI) and consider the partial type
Z(x) = {xx” - u2 =o, P&&X), n a 2).
Claim. Z is realized in MZ.
Assume the claim is true and let y, realize 2’. First observe that X4 - Ui is irreducible over Ai and that the induced valuation on Ai is completely determined, namely
v(e, + eryi + * * * + eq_lyf-l) = min V(f?jY{).
So we get an isomorphism of valued fields fN : A,( y,)* A*( y2) extending f and sending y, onto y,. Let us see that f” preserves the P,,. Let xl E Al(yl), x2 = f”(xl).
Substructures and uniform elimination for p-adic fields 15
We have v(x,d,y,-‘) = 0 for some di E Al and some 0 ~j < q. Let d2 =f(d,); then v(x2dZy;j) = 0. There exists I, cAI such that v(&) = 0 and A,x,d,y;jc Mi”. Let A2 =f(A,); then A,x,d,y,‘E Mi” (cf. (2.2)). Hence x1 E Mi” iff A,d,y;‘E Mi” iff hldlpi E Mi” iff &d&i E M;” iff L,d,y;‘E M;” iff x2 E M;” and we are done. By (2.2) the Sol,, are again taken care of by the residue fields. So f” is an 9?‘-isomorphism.
Proof of the Claim. Since we are in equal characteristic 0, the number of q-th roots of 1 in the Mi is decided in the residue fields and by (2.2) has to be the same. If there is only one q-th root of 1, there is only one choice for yz and nothing to prove since, then, x EM;” iff x4 E Minq and (p,,~~)~ = p,4al cA1, etc. So let I; be a primitive q-th root of 1 in M2 and b E M2 be such that bq = u2. Suppose the claim false. Then there are n,,, . . . , IZ~_~ such that p,,,bcj 4 M;‘?. Let n = Icm(nj), then p,,bcj 4 Mi” for all j (pclpni E M;“,). But pnyl E Mi” implies
p$z, = (p,b)q E M;“. So (p,bx”)q = 1 for some x E M2 and p,bl;’ E M;” for some j, contradiction. This completes the proof of the claim and (2.3).
(2.4) So we now have Al henselian, res A, = res Ml and val A, pure in val Ml. Since val Ml is a Z-group and val A, has the same unit, this implies that val A, is a h-group as well. Hence by Ax-Kochen-Ershov Al =5 Ml as valued fields, so that clearly &I < Jui. Now, since .& is w,-saturated, it follows that f extends to an embedding J& L, 4. 0
The argument related to uniqueness of p-adic closures lies in part (2.3) of the preceding proof.
Theorem 3.8 can be deduced from the Main Theorem in [27] as follows. The setting is a 2-sorted language for valued fields, with additional sorts Rk, k E w, for the residue rings V/(tk+‘). By Variant 4.5 (ibid.) with li- = v(t), q = t, E,, = {U*,jt 11 sj s 6’(n)} and C = 0, there is a primitive recursive procedure to eliminate the base field quantifiers. To get rid of the Rk sorts for k >O, we replace the Rk-variables (terms) by RO-variables (terms), using the basic idea that an element of Rk is essentially determined by a finite sum y. + ylt + * * - + yktk, v(y,) = 0. The problem is to do this in T’. Now for any non-principal ultrafilter 9 on the set of primes the ultraproducts U QP/9 and fl FP((T))/9 are elementarily equivalent as valued fields. So, given a polynomial f E Z[X,, . . . , X,] and k > 0 there is a bound N(f) and polynomials g E Z[&, . . . , Y,], x = (xl, . . . , &), such that for all p > N(f) the condition Rk bf(x,, . . . , x,) = 0 is equivalent (in T’) to a finite set of conditions RoLg(y,, . . . , yn) = 0, where, e.g., if xi 3 1 Zjt’ mod tk+‘, then ~j = Yij, those equations being obtained by identifying Rk with Ro[T]/(Tk+‘), T transcendental over Ro, char R. = 0. This N(f) can be obtained primitive recursively as in [lo, Section 51, or even explicitly by [6]. In this way we can replace the Rk-quantifiers by Ro-quantifiers. Clearly this procedure is still primitive recursive. This brings us into conditions similar to those of Theorem 4.2 (ibid.) allowing the transfer of (primitive recursive) E.Q.
16 L. Bdair
from (the theory of. . .) the value group and the residue field to the whole structure. Hence we get Theorem 3.8 from axiom (7) of T’ together with standard E.Q. for Z-groups on the one hand, and the Sol,, predicates together with Lemma 3.2 on the other hand. It is well known that the theory of H-groups admits primitive recursive E.Q. It is also known that a primitive recursive E.Q. procedure exists for the elementary theory of finite fields, as given by [16], and that it can be put in the formalism of Lemma 3.2. Putting everything together we conclude that T’ admits primitive recursive elimination of quantifiers in 2’.
One sees that a suitable version of Theorem 3.8 also holds for finite extensions of Qp of a given degree d. The same kind of argument applies for the index of PA,
etc. However there is no uniform bound for [K’ : PA] for arbitrarily large finite extensions K/Q, for fixed II, even if p is fixed, so our method fails for such classes of local fields.
Example 3.9. Consider the index of P; in Q2(21’2), . . . , Q2(21’2”), . . . . Let & = 2112”. The valuation ring of Q2(4 is Z2[a,J and it suffices to look at the number of square roots of 1 mod CLI?+~. Now, e.g., consider Q2(a2). A typical element of Z2(21’4)/(2 * 21’4) looks like C”, iEi2i’4, Ai E (0, l}, and when it is squared the parameters &, A4 disappear. So there are at least 22 square roots of 1. Similarly, there are at least 2” square roots of 1 in Z2[41(&““+‘). Hence [@(an) : P,] 2 2*+l. ,A similar argument works for unramified extensions and any other p.
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